the diameter of the circle. His calculation is based on two close approximations (an upper and a lower) to the value of √3, that being the cotangent of the angle of 30°, from which he begins to work. He assumes as known that 265/153 < √3 < 1351/780. In the text, as we have it, only the results of the steps in the calculation are given, but they involve the finding of approximations to the square roots of several large numbers: thus 11721⁄8 is given as the approximate value of √(137394333⁄64), 3013¾ as that of √(9082321) and 18389⁄11 as that of √(3380929). In this way Archimedes arrives at 14688 / 4673½ as the ratio of the perimeter of the circumscribed polygon of 96 sides to the diameter of the circle; this is the figure which he rounds up into 31⁄7. The corresponding figure for the inscribed polygon is 6336 / 2017¼, which, he says, is > 310⁄71. This example shows how little the Greeks were embarrassed in arithmetical calculations by their alphabetical system of numerals.

On Conoids and Spheroids.

The preface addressed to Dositheus shows, as we may also infer from internal evidence, that the whole of this book also was original. Archimedes first explains what his conoids and spheroids are, and then, after each description, states the main results which it is the aim of the treatise to prove. The conoids are two. The first is the right-angled conoid, a name adapted from the old name (“section of a right-angled cone”) for a parabola; this conoid is therefore a paraboloid of revolution. The second is the obtuse-angled conoid, which is a hyperboloid of revolution described by the revolution of a hyperbola (a “section of an obtuse-angled cone”) about its transverse axis. The spheroids are two, being the solids of revolution described by the revolution of an ellipse (a “section of an acute-angled cone”) about (1) its major axis and (2) its minor axis; the first is called the “oblong” (or oblate) spheroid, the second the “flat” (or prolate) spheroid. As the